MATHCOUNTS - WeeblyMATHCOUNTS Founding Sponsors: National Society of Professional Engineers,...

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2013 State Competition Sprint Round Problems 1–30 Total Correct Scorer’s Initials Copyright MATHCOUNTS, Inc. 2013. All rights reserved. MATHCOUNTS ® Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. If you are wearing a calculator wrist watch, please give it to your proctor now. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers. In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted. 03-S13SPR HONOR PLEDGE I pledge to uphold the highest principles of honesty and integrity as a Mathlete ® . I will neither give nor accept unauthorized assistance of any kind. I will not copy another’s work and submit it as my own. I understand that any competitor found to be in violation of this honor pledge is subject to disqualification. Signature _____________________________________________ Date ___________ Printed Name __________________________________________________________ School _______________________________________________________________ Chapter ______________________________________________________________ National Sponsors Raytheon Company * Northrop Grumman Foundation * U.S. Department of Defense * National Society of Professional Engineers * CNA Foundation * ConocoPhillips * Texas Instruments Incorporated * 3M Foundation * Art of Problem Solving * NextThought

Transcript of MATHCOUNTS - WeeblyMATHCOUNTS Founding Sponsors: National Society of Professional Engineers,...

  • 2013State Competition

    Sprint RoundProblems 1–30

    Total Correct Scorer’s Initials

    Copyright MATHCOUNTS, Inc. 2013. All rights reserved.

    MATHCOUNTS ®

    Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation

    DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

    This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. If you are wearing a calculator wrist watch, please give it to your proctor now. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers.

    In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted.

    03-S13SPR

    HONOR PLEDGE I pledge to uphold the highest principles of honesty and integrity as a Mathlete®. I will neither give nor accept unauthorized assistance of any kind. I will not copy another’s work and submit it as my own. I understand that any competitor found to be in violation of this honor pledge is subject to disqualification.

    Signature _____________________________________________ Date ___________

    Printed Name __________________________________________________________

    School _______________________________________________________________

    Chapter ______________________________________________________________

    National Sponsors Raytheon Company * Northrop Grumman Foundation * U.S. Department of Defense * National Society of Professional Engineers * CNA Foundation * ConocoPhillips * Texas Instruments Incorporated * 3M Foundation * Art of Problem Solving * NextThought

  • Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

    1. _____________

    2. _____________

    3. _____________

    4. _____________

    5. _____________

    Ifthesumoftwonumbersis4andtheirdifferenceis2,whatistheirproduct?

    MaryandAnnliveatoppositeendsofthesameroad.Theyplantoleavehomeatthesametimeandridetheirbikestomeetsomewherebetweenthetwohouses.At11:00a.m.Maryhastraveledhalfofthedistancebetweentheirhouses.Annisridingmoreslowlyandhascoveredonly

    38ofthedistance

    betweenthehouses.Theyarestillonemileapart.Howmanymilesapartaretheirhouses?

    Abagcontains8bluemarbles,4redmarblesand3greenmarbles.Inasingledraw,whatistheprobabilityofnotdrawingagreenmarble?Expressyouranswerasacommonfraction.

    Thearithmeticmeanof11numbersis78.If1issubtractedfromthefirst,2issubtractedfromthesecond,3issubtractedfromthethird,andsoforth,until11issubtractedfromtheeleventh,whatisthearithmeticmeanofthe11resultingnumbers?

    Anoptometristhasthislogoonhisstorefront.Thecentercirclehasarea36πin2,anditistangenttoeachcrescentatitswidestpoint(AandB).TheshortestdistancefromAtotheoutercircleis

    13 thediameterofthesmaller

    circle.Whatistheareaofthelargercircle?Expressyouranswerintermsofπ.

    in2

    miles

    A

    B

  • Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

    Excludingsalestax,howmuchwillDorissavewhenshebuysaDVDoriginallypricedat$12.00andnowonsalefor20%off ?

    Whatisthevalueof2 2444 111

    444 111−− ?

    Theproductofthedigitsofpositiveintegern is20,andthesumofthedigitsis13.Whatisthesmallestpossiblevalueofn?

    QuadrilateralABCDisasquarewithBC=12cm.BOC andDOC aresemicircles.Intermsofπ,whatistheareaoftheshadedregion?

    Realnumbersaandbsatisfytheequation 2 4 3 15 5− +

    + =a a b .Whatisthe

    valueofa−b?Expressyouranswerasacommonfraction.

    Ifthepoint(x,x)isequidistantfrom(−2,5)and(3,−2),whatisthevalueofx?

    6. _____________

    7. _____________

    8. _____________

    9. _____________

    10._____________

    11._____________

    cm2

    O

    C

    B

    D

    A

    12

    $

  • Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

    12._____________

    13._____________

    14._____________

    15._____________

    16._____________

    17._____________

    18._____________

    Inabagofmarbles,25 ofthemarblesarered,

    310 ofthemarblesare

    whiteand110 ofthemarblesareblue.Iftheremaining10marbles

    aregreen,howmanymarblesareinthebag?

    Iftis40%greaterthanp,andpis40%lessthan600,whatisthevalueoft–p?

    Howmanywayscanallsixnumbersintheset{4,3,2,12,1,6}beorderedsothatacomesbeforebwheneveraisadivisorofb?

    Whatistheunitsdigitoftheproduct723×8105×318?

    If4(a–3)–2(b+5)=14and5b–a=0,whatisthevalueofa+b?

    ThetwoconesshownhaveparallelbasesandcommonapexT.TW=32m,WV=8mandZY=5m.WhatisthevolumeofthefrustumwithcircleWandcircleZasitsbases?Expressyouranswerintermsofπ.

    Acoinisflippeduntilithaseitherlandedheadstwotimesortailstwotimes,notnecessarilyinarow.Ifthefirstfliplandsheads,whatistheprobabilitythatasecondheadoccursbeforetwotails?Expressyouranswerasacommonfraction.

    marbles

    m3

    ways

    T

    YZX

    VWU

  • Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

    19._____________

    20._____________

    21._____________

    22._____________

    23._____________

    24._____________

    Theproductoftwoconsecutiveintegersisfivemorethantheirsum.Whatisthesmallestpossiblesumoftwosuchconsecutiveintegers?

    Fournickels,onepennyandonedimeweredividedamongthreepiggybankssothateachbankreceivedtwocoins.Labelsindicatingtheamountineachbankweremade(6cents,10centsand15cents),butwhenthelabelswereputonthebanks,nobankhadthecorrectlabelattached.Sorayashookthepiggybanklabeledas15cents,andoutfellapenny.Whatwastheactualcombinedvalueofthetwocoinscontainedinthepiggybankthatwaslabeled6cents?

    Supposethe9×9multiplicationgrid,shownhere,werefilledincompletely.Whatwouldbethesumofthe81products?

    Insomelanguages,everyconsonantmustbefollowedbyavowel.Howmanyseven-letter‟words”canbemadefromtheHawaiianwordMAKAALAifeachconsonantmustbefollowedbyavowel?

    Iff(x)=3x2,whatisthex-coordinateofthepointofintersectionofthegraphsofy=f(x)andy=f(x−4)?

    InisoscelestrapezoidABCD,shownhere,AB=4unitsandCD=10units.PointsEandFareonCD with BE paralleltoAD andAF paralleltoBC .AF andBE intersectatpointG.WhatistheratiooftheareaoftriangleEFGtotheareaoftrapezoidABCD?Expressyouranswerasacommonfraction.

    ‟words”

    A B

    CDE F

    G

    cents

    × 1 2 3 4 5 6 7 8 9123456789

    14

    12

    40

    6¢ 10¢

    15¢

  • Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

    25._____________

    26._____________

    27._____________

    28._____________

    29._____________

    30._____________

    Thesumoffiveconsecutive,positiveevenintegersisaperfectsquare.Whatisthesmallestpossibleintegerthatcouldbetheleastofthesefiveintegers?

    If123+125+127+129+12x=1011102,whatisthevalueofx,thebaseofthefifthterm?

    Aboxcontains r redballsandggreenballs.Whenrmoreredballsareaddedtothebox,theprobabilityofdrawingaredballatrandomfromtheboxincreasesby25%.Whatwastheprobabilityofrandomlydrawingaredballfromtheboxoriginally?Expressyouranswerasacommonfraction.

    ThegameofConnexcontainsone4-unitpiece,twoidentical3-unitpieces,threeidentical2-unitpiecesandfouridentical1-unitpieces.Howmanydifferentarrangementsofpieceswillmakea10-unitsegment?The10-unitsegmentsconsistingofthepieces4-3-2-1and1-2-3-4aretwosucharrangementstoinclude.

    Insquareunits,whatisthelargestpossibleareaarectangleinscribedinthetriangleshownherecanhave?

    AlinesegmentwithendpointsA(3,1)andB(2,4)isrotatedaboutapointintheplanesothatitsendpointsaremovedtoA' (4,2)andB' (7,3),respectively.Whatarethecoordinatesofthecenterofrotation?Expressyouranswerasanorderedpair.

    10 17

    21

    units2

    (,)

    arrange-ments

    2013 Cover State Sprint.pdfBlank page.pdf2013 State Sprint - FINAL.pdf